DSP-Lec 04-FIR Filtering and Convolution

Chapter p FIR filtering and Convolution Ha Hoang Kha, Ph.D.Click to edit Master subtitle style Ho Chi Minh City University of Technology @ Email: hhkha@hcmut.edu.vn Content ™ Block p processingg methods ‰ Convolution: direct form, convolution table ‰ Convolution: LTI form,, LTI table ‰ Matrix form ‰ Flip-and-slide form ‰ Overlap-add block convolution method ™ Sample processing methods ‰ FIR filtering in direct form Ha H Kha FIR Filtering and Convolution Introduction ™ Block processing methods: data are collected and processed in blocks ‰ FIR filtering of finite finite-duration duration signals by convolution ‰ Fast convolution of long signals which are broken up in short segments ‰ DFT/FFT spectrum computations ‰ Speech analysis and synthesis ‰ Image processing ™ Sample processing methods: the data are processed one at a timewith each input sample being subject to a DSP algorithm which transforms it into an output sample ‰ Real-time applications ‰ Digitall audio d effects ff processing ‰ Digital control systems ‰ Adaptive signal processing Ha H Kha FIR Filtering and Convolution Block Processing method ™ The collected signal samples x(n), n=0, 1,…, L-1, can be thought as a block: x=[x [ 0, x1, …, xL-1] The duration of the data record in second: TL=LT ™ Consider a casual FIR filter of order M with impulse response: h=[h0, h1, …, hM] The length (the number of filter coefficients): Lh=M+1 Ha H Kha FIR Filtering and Convolution 11.1 Direct form ™ The convolution in the direct form: y ( n) = ∑ h( m) x ( n − m) m ™ For DSP implementation, p , we must determine ‰ The range of values of the output index n ‰ The precise range of summation in m ™ Find index n: index of h(m) Ỉ 0≤m≤M index of x(n-m) x(n m) Ỉ 0≤n-m≤L-1 0≤n m≤L Ỉ ≤ m ≤ n ≤m+L-1 ≤ M+L-1 ≤ n ≤ M + L −1 ™ Lx=L input samples which is processed by the filter with order M M=L Lx + M yield the output signal y(n) of length L y = L + M Ha H Kha FIR Filtering and Convolution 1Direct form ™ Find index m: index of x(n-m) ( ) index of h(m) Ỉ 0≤m≤M Ỉ 0≤n-m≤L-1 Ỉ n+L-1≤ m ≤ n max ( 0, n − L + 1) ≤ m ≤ ( M, n ) ™ The direct form of convolution is given as follows: y ( n) = min( M , n ) ∑ h( m) x ( n − m) = h ∗ x m = max(0, (0 n − L +1) with ≤ n ≤ M + L − ™ Thus,, y is longer g than the input p x byy M samples p This property p p y follows from the fact that a filter of order M has memory M and keeps each input sample inside it for M time units Ha H Kha FIR Filtering and Convolution Example ™ Consider the case of an order-3 filter and a length of 5-input signal Find the output o t ? h=[h0, h1, h2, h3] x=[x0, x1, x2, x3, x4 ] yy=h*x=[y h x [y0, y1, y2, y3, y4 , y5, y6, y7 ] Ha H Kha FIR Filtering and Convolution 1.2 Convolution table ™ It can be observed that y ( n) = ∑ h(i) x( j ) i, j i+ j =n ™ Convolution table ™ The convolution table is convenient for quick calculation b h by hand d because b iit displays all required operations compactly Ha H Kha FIR Filtering and Convolution Example ™ Calculate the convolution of the following filter and input signals? h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1] ™ Solution: S l i sum of the values along anti-diagonal line yields the output y: y=[1, =[1 3, 3, 5, 3, 7, 4, 3, 3, 0, 1] Note that there are Ly=L+M=8+3=11 output samples Ha H Kha FIR Filtering and Convolution 1.3 LTI Form ™ LTI form of convolution: y ( n ) = ∑ x ( m) h ( n − m) m ™ Consider the filter h=[h0, h1, h2, h3] and the input signal x=[x0, x1, x2, x3, x4 ] Then, the output is given by y (n) = x0 h(n) + x1h(n − 1) + x2 h(n − 2) + x3 h(n − 3) + x4 h(n − 4) ™ We can represent the input and output signals as blocks: Ha H Kha 10 FIR Filtering and Convolution 1.3 Matrix Form ™ It can be b observed b d th thatt H has h th the same entry t along l each h diagonal di l Such a matrix is known as Toeplitz matrix ™ Matrix representations of convolution are very useful in some applications: ‰ Image processing ‰ Advanced DSP methods such as parametric spectrum estimation and adaptive filtering Ha H Kha 14 FIR Filtering and Convolution Example ™ Using the matrix form to calculate the convolution of the following filter and input signals? h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1] ™ Solution: since Lx=8, M=3 Ỉ Ly=Lx+M=11, the filter matrix is 11x8 dimensional Ha H Kha 15 FIR Filtering and Convolution 1.4 Flip-and-slide form ™ The output at time n is given by yn = h0 xn + h1 xn −1 + + hM xn − M ™ Flip-and-slide li d lid fform off convolution l i ™ The flip-and-slide form shows clearly the input-on and input-off transient and steady-state steady state behavior of a filter Ha H Kha 16 FIR Filtering and Convolution 1.5 Transient and steady-state behavior M ™ From LTI convolution: y(n) = ∑h(m)x(n − m) = h0 xn + h1xn−1 + + hM xn−M m=0 ™ The output is divided into subranges: ™ Transient T i andd steady-state d fil outputs: filter Ha H Kha 17 FIR Filtering and Convolution 1.6 Overlap-add block convolution method ™ As the input signal is infinite or extremely large, a practical approach is to divide the longg input p into contiguous g non-overlapping pp g blocks of manageable length, say L samples ™ Overlap Overlap-add add block convolution method: Ha H Kha 18 FIR Filtering and Convolution Example ™ Using the overlap-add method of block convolution with each bock length L=3, calculate the convolution of the following filter and input signals? h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1] ™ Solution: The input is divided into block of length L L=33 The output of each block is found by the convolution table: Ha H Kha 19 FIR Filtering and Convolution Example ™ The output of each block is given by ™ Following F ll i from f time i invariant, i i aligning li i the h output blocks bl k according di to theirs absolute timings and adding them up gives the final results: Ha H Kha 20 FIR Filtering and Convolution Sample processing methods ™ The direct form convolution for an FIR filter of order M is given by ™ Introduce the internal states Sample p p processingg algorithm g Fig: Direct form realization off Mth M h order d fil filter Ha H Kha ™ Sample processing methods convenient for real-time real time applications 21 are FIR Filtering and Convolution Example ™ Consider the filter and input given by Using the sample processing algorithm to compute the output and show the input-off transients Ha H Kha 22 FIR Filtering and Convolution Example Ha H Kha 23 FIR Filtering and Convolution Example Ha H Kha 24 FIR Filtering and Convolution Hardware realizations ™ The FIR filtering algorithm can be realized in hardware using DSP chips, for example the Texas Instrument TMS320C25 ™ MAC: Multiplier Accumulator Ha H Kha 25 FIR Filtering and Convolution ... Kha 22 FIR Filtering and Convolution Example Ha H Kha 23 FIR Filtering and Convolution Example Ha H Kha 24 FIR Filtering and Convolution Hardware realizations ™ The FIR filtering algorithm can... FIR Filtering and Convolution 1.4 Flip -and- slide form ™ The output at time n is given by yn = h0 xn + h1 xn −1 + + hM xn − M ™ Flip -and- slide li d lid fform off convolution l i ™ The flip -and- slide... the input-on and input-off transient and steady-state steady state behavior of a filter Ha H Kha 16 FIR Filtering and Convolution 1.5 Transient and steady-state behavior M ™ From LTI convolution: